# Use the Mean Value Theorem to proof the inequality

Need to prove that $57/8 < \sqrt{51}$ using the mean value theorem. The hint given in the question is to use the $\sqrt{x}$ function. I have used the interval $51 < c < 64$.

Therefore I have: $f(64)-f(51)=f'(c)(64-51)=8-\sqrt{51}=13/2\sqrt{c}$

What I have trouble here is isolating $\sqrt{51}$ into an inequality. Have I used a correct interval to use the Mean Value Theorem? Is there a trick I'm missing? Thanks in advance.

Your method is correct. Choose the interval to be $[49, 51]$. That would work.
Hint: Use the fact that $51<c<64$ to get $\frac{1}{2\sqrt 64}< \frac{1}{2\sqrt c}<\frac{1}{2\sqrt 51}$ and the result follows.